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1、May 19, 2003,73-347 Game Theory-Lecture 1,1,Static (or Simultaneous-Move) Games of Complete Information,Introduction to Games Normal (or Strategic) Form Representation,May 19, 2003,73-347 Game Theory-Lecture 1,2,Outline of Static Games of Complete Information,Introduction to games Normal-form (or st
2、rategic-form) representation Iterated elimination of strictly dominated strategies Nash equilibrium Review of concave functions, optimization Applications of Nash equilibrium Mixed strategy Nash equilibrium,May 19, 2003,73-347 Game Theory-Lecture 1,3,Agenda,What is game theory Examples Prisoners dil
3、emma The battle of the sexes Matching pennies Static (or simultaneous-move) games of complete information Normal-form or strategic-form representation,May 19, 2003,73-347 Game Theory-Lecture 1,4,What is game theory?,We focus on games where: There are at least two rational players Each player has mor
4、e than one choices The outcome depends on the strategies chosen by all players; there is strategic interaction Example: Six people go to a restaurant. Each person pays his/her own meal a simple decision problem Before the meal, every person agrees to split the bill evenly among them a game,May 19, 2
5、003,73-347 Game Theory-Lecture 1,5,What is game theory?,Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) who behave strategically Game theory has applications Economics Politics etc.,May 19, 2003,73-347 Game Theory-Lecture 1,6,Classic Example
6、: Prisoners Dilemma,Two suspects held in separate cells are charged with a major crime. However, there is not enough evidence. Both suspects are told the following policy: If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. If both confess then bot
7、h will be sentenced to jail for six months. If one confesses but the other does not, then the confessor will be released but the other will be sentenced to jail for nine months.,May 19, 2003,73-347 Game Theory-Lecture 1,7,Example: The battle of the sexes,At the separate workplaces, Chris and Pat mus
8、t choose to attend either an opera or a prize fight in the evening. Both Chris and Pat know the following: Both would like to spend the evening together. But Chris prefers the opera. Pat prefers the prize fight.,May 19, 2003,73-347 Game Theory-Lecture 1,8,Example: Matching pennies,Each of the two pl
9、ayers has a penny. Two players must simultaneously choose whether to show the Head or the Tail. Both players know the following rules: If two pennies match (both heads or both tails) then player 2 wins player 1s penny. Otherwise, player 1 wins player 2s penny.,May 19, 2003,73-347 Game Theory-Lecture
10、 1,9,Static (or simultaneous-move) games of complete information,A set of players (at least two players) For each player, a set of strategies/actions Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies,Player
11、 1, Player 2, . Player n S1 S2 . Sn ui(s1, s2, .sn), for all s1S1, s2S2, . snSn.,A static (or simultaneous-move) game consists of:,May 19, 2003,73-347 Game Theory-Lecture 1,10,Static (or simultaneous-move) games of complete information,Simultaneous-move Each player chooses his/her strategy without k
12、nowledge of others choices. Complete information Each players strategies and payoff function are common knowledge among all the players. Assumptions on the players Rationality Players aim to maximize their payoffs Players are perfect calculators Each player knows that other players are rational,May
13、19, 2003,73-347 Game Theory-Lecture 1,11,Static (or simultaneous-move) games of complete information,The players cooperate? No. Only noncooperative games The timing Each player i chooses his/her strategy si without knowledge of others choices. Then each player i receives his/her payoff ui(s1, s2, .,
14、 sn). The game ends.,May 19, 2003,73-347 Game Theory-Lecture 1,12,Definition: normal-form or strategic-form representation,The normal-form (or strategic-form) representation of a game G specifies: A finite set of players 1, 2, ., n, players strategy spaces S1 S2 . Sn and their payoff functions u1 u2
15、 . un where ui : S1 S2 . SnR.,May 19, 2003,73-347 Game Theory-Lecture 1,13,Normal-form representation: 2-player game,Bi-matrix representation 2 players: Player 1 and Player 2 Each player has a finite number of strategies Example: S1=s11, s12, s13 S2=s21, s22,May 19, 2003,73-347 Game Theory-Lecture 1
16、,14,Classic example: Prisoners Dilemma: normal-form representation,Set of players: Prisoner 1, Prisoner 2 Sets of strategies: S1 = S2 = Mum, Confess Payoff functions: u1(M, M)=-1, u1(M, C)=-9, u1(C, M)=0, u1(C, C)=-6; u2(M, M)=-1, u2(M, C)=0, u2(C, M)=-9, u2(C, C)=-6,Payoffs,May 19, 2003,73-347 Game Theory-Lecture 1,15,Example: The battle of the sexes,Normal (or strategic) form representation: Set of players: Chris, Pat (=Play
